Section 12.2 Question 1

How is the second derivative calculated?

The second derivative is the derivative of the derivative. To help identify this process, the first time we take a derivative of a function we call it the first derivative. The first derivative, as we have seen earlier, is symbolized in several different ways. If we take the first derivative of a function y = f (x), the first derivative is written as12_2_1_1

Several notations for the second derivative are used.

Any of the following notations may be used to write the second derivative of a function :12_2_1_2

We can take the derivative of the first derivative by applying the rules for derivatives to the first derivative. If f (x) = 5x4 – 7x2 +2x – 1, then we can apply the Sum / Difference, Product with a Constant, and  Power Rules for Derivatives to yield the first derivative12_2_1_3

If we use these rules again on the first derivative, we get the second derivative,12_2_1_4

The process for finding the derivative is the same as we have used in Chapter 11. The only difference is the starting point. When finding the first derivative of a function, we take the derivative of the function. For the second derivative, we take the derivative of the first derivative.

Example 1      Calculate the Second Derivative

Let  f (x) = x3 -4x2 + 6x +12. Find the second derivative ″(x).

Solution The first derivative of  is found by applying rules for derivatives,12_2_1_5

To find ″(x), take the derivative of f ′(x) = 3x2 – 8x +6:


The second derivative is  f ″(x) = 6x – 8.

Example 2     Calculate the Second Derivative


Find the second derivative 12_2_1_8

Solution The first derivative is found with the Product Rule for Derivatives using the factors


The first derivative is


Apply the product rule again to find the second derivative with the factors


The second derivative is


The second derivative is 12_2_1_13

Example 3      Calculate the Second Derivative

Let 12_2_1_14Find the second derivative g″(x).

Solution Use the Quotient Rule for Derivatives to find the first derivative with


The first derivative is


To compute the second derivative, take the derivative of the first derivative with the Quotient Rule for Derivatives,


The second derivative is


The second derivative expression can be simplified further by factoring the numerator:


The second derivative is 12_2_1_20

Goto How are other higher derivatives calculated?

Goto Beginning of Section 12.2